What is the purpose of fermenting in a secondary, and tertiary fermenter? Is it simple to filter the beer so there are not things floating around in the beer?

There is no purpose. Stick with primary only. In 10-15 days (dependant on recipe) transfer to kegs and age.

Even if not kegging, transfer to a bottleing bucket with priming sugar and bottle. Essentially, as long as you have healthy yeast, you can leave the beer on the yeast for 3 weeks for sure, possibly 4 to 6 weeks in most cases, without any negative effects. Transfering to a secondary is just one more chance to infect the beer in my opinion.

In the "old days" when yeast was not the quality it is today, and just taped to the top of and exract can and left in the warehouse for months and months... the yeast was on it's last legs when pitched into the wort, so getting it off the yeast was important. Now all that is much better!

The practice of secondary fermentation has fallen out of favor with homebrewers and is considered redundant as far as I can tell. I just go to keg or bottle once the attenuative phase has passed. That can be as soon as 5 days or up to 10 or more.

I've done it a few times, back in the 90's and when dry-hopping. It's not about fermentation cause it's 99% done already, but to let more yeast and "floaties" settle out, or to add fruit, herbs, spices or whatnot. Believe me if one uses a yeast with high flocculation characteristics it'll be clear in the primary and certainly no need for additional clarification techniques as cumbersome as a secondary vessel. It can all be done in primary for that matter...

Logged

The first principle is that you must not fool yourself, and you are the easiest person to fool. -Richard P. Feynman

Laws are spider-webs, which catch the little flies, but cannot hold the big ones. -Anacharsis

I secondary, when I have no more primaries left and happen to be brewing. Buckets are easier to clean than carboys. Sometimes my drinking outpaces my brewing - other times my brewing outpaces my drinking.

Most low gravity beers don't need a secondary and it is arguable wether or not there is benefit or not. Personally, I keg everything (and when I bottle I bottle form the keg) so I never secondary in a carboy except in the case of sour beers and the rare occasion I use fruit. Some fruit gets so chunky that you may have to use a tertiary fermenter to leave everything behind. But that is a rarity.

A secondary is essentially what breweries call a conditioning tank. But breweries tend to have faster primaries, so moving the beer off gives them the ability to use the fermenter again. In this case, the secondary is important for clarification and for maturing the beer (reducing acetaldehyde, diacetyl, and other green beer flavors that can result from premature separation from the yeast).

Homebrewers typically leave their beer in the primary longer, until the yeast flocs out. In that case, you've pretty much used the primary as a secondary already. Taste your beer and see if it has any green flavors that need to be dealt with; if so, leave it on the yeast.

A secondary is useful for homebrewers if you're going to be doing additional fiddling with the beer that requires leaving it around at warmer temperatures, like dry hopping, oaking, adding other flavors, etc. Basically, you're moving the beer off the bulk of the yeast and trub to avoid picking up off flavors.

If your beer has dropped bright in the primary and you're ready to package, just do it. If you need to hold the beer for awhile (especially if warm), move it off the yeast to avoid autolysis flavors. It's about the same amount of work to transfer to a secondary as it is to keg, so the only reason I'd do this is if I didn't have kegs available (all full, need cleaning, etc.). If you do use a secondary, blow CO2 into the carboy first to avoid oxygen pickup during racking.

Filtering is a pretty abusive thing to do to beer, so I'd try fining and time first. If you rack unfiltered into a keg, just give it time and be prepared to toss the first pint or two you pull. You can fine in a secondary or in a keg as well.

I secondary, maybe I'm outdated, but I frequently get caught up with things and leave in primary for 2-3 weeks and still have a significant fallout in my secondary. May very well be from trying to siphon too much and not waste any beer. To each their own. Do what works for you.